Expression of type Forall

from the theory of proveit.physics.quantum.circuits

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit.core_expr_types.expr_arrays import A11_to_Akl
from proveit.logic import Equals, Forall
from proveit.physics.quantum.circuits import circuit__u_Akl_v, circuit__u_Akl_v__psi_m
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [A11_to_Akl], instance_expr = Equals(Prob(circuit__u_Akl_v), Prob(circuit__u_Akl_v__psi_m)).with_wrapping_at(2)).with_wrapping()

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\begin{array}{l}\forall_{A_{1, 1}, A_{1, 2}, \ldots, A_{1, l}, A_{2, 1}, A_{2, 2}, \ldots, A_{2, l}, \ldots\ldots, A_{k, 1}, A_{k, 2}, \ldots, A_{k, l}}~\\
\left(\begin{array}{c} \begin{array}{l} \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\multiqin{3}{\lvert u \rangle} & \gate{A_{1, 1}} \qwx[1] & \gate{A_{2, 1}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 1}} \qwx[1] & \multiqout{3}{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, 2}} \qwx[1] & \gate{A_{2, 2}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 2}} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, l}} & \gate{A_{2, l}} & \gate{\cdots} & \gate{A_{k, l}} & \ghostqout{\lvert v \rangle}
} \end{array}\right) =  \\ \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\multiqin{3}{\lvert u \rangle} & \gate{A_{1, 1}} \qwx[1] & \gate{A_{2, 1}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 1}} \qwx[1] & \multiqout{3}{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, 2}} \qwx[1] & \gate{A_{2, 2}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 2}} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, l}} & \gate{A_{2, l}} & \gate{\cdots} & \gate{A_{k, l}} & \ghostqout{\lvert v \rangle} \\
\qin{\lvert \psi \rangle} & { /^{m} } \qw & { /^{m} } \qw & \gate{\cdots} & { /^{m} } \qw & \qout{\lvert \psi \rangle}
} \end{array}\right) \end{array} \end{array}\right)\end{array}

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 4 body: 5
4	ExprTuple	6
5	Operation	operator: 7 operands: 8
6	ExprRange	lambda_map: 9 start_index: 85 end_index: 30
7	Literal
8	ExprTuple	10, 11
9	Lambda	parameter: 75 body: 42
10	Operation	operator: 13 operand: 15
11	Operation	operator: 13 operand: 16
12	ExprTuple	15
13	Literal
14	ExprTuple	16
15	Operation	operator: 18 operands: 17
16	Operation	operator: 18 operands: 19
17	ExprTuple	20, 21, 22
18	Literal
19	ExprTuple	23, 24, 25
20	ExprTuple	27
21	ExprRange	lambda_map: 26 start_index: 85 end_index: 30
22	ExprTuple	31
23	ExprTuple	27, 28
24	ExprRange	lambda_map: 29 start_index: 85 end_index: 30
25	ExprTuple	31, 32
26	Lambda	parameter: 75 body: 33
27	ExprRange	lambda_map: 34 start_index: 85 end_index: 86
28	ExprRange	lambda_map: 35 start_index: 85 end_index: 87
29	Lambda	parameter: 75 body: 37
30	Variable
31	ExprRange	lambda_map: 38 start_index: 85 end_index: 86
32	ExprRange	lambda_map: 39 start_index: 85 end_index: 87
33	ExprTuple	42
34	Lambda	parameter: 79 body: 40
35	Lambda	parameter: 79 body: 41
36	ExprTuple	75
37	ExprTuple	42, 43
38	Lambda	parameter: 79 body: 44
39	Lambda	parameter: 79 body: 45
40	Operation	operator: 51 operands: 46
41	Operation	operator: 51 operands: 47
42	ExprRange	lambda_map: 48 start_index: 85 end_index: 86
43	ExprRange	lambda_map: 49 start_index: 85 end_index: 87
44	Operation	operator: 51 operands: 50
45	Operation	operator: 51 operands: 52
46	NamedExprs	element: 53 targets: 59
47	NamedExprs	element: 54 targets: 61
48	Lambda	parameter: 79 body: 55
49	Lambda	parameter: 79 body: 57
50	NamedExprs	element: 58 targets: 59
51	Literal
52	NamedExprs	element: 60 targets: 61
53	Operation	operator: 63 operands: 62
54	Operation	operator: 63 operands: 71
55	IndexedVar	variable: 64 indices: 65
56	ExprTuple	79
57	Operation	operator: 66 operands: 67
58	Operation	operator: 70 operands: 68
59	Operation	operator: 72 operands: 69
60	Operation	operator: 70 operands: 71
61	Operation	operator: 72 operands: 73
62	NamedExprs	state: 74 part: 79
63	Literal
64	Variable
65	ExprTuple	75, 79
66	Literal
67	NamedExprs	operation: 76
68	NamedExprs	state: 77 part: 79
69	ExprTuple	85, 86
70	Literal
71	NamedExprs	state: 78 part: 79
72	Literal
73	ExprTuple	80, 81
74	Variable
75	Variable
76	Literal
77	Variable
78	Variable
79	Variable
80	Operation	operator: 83 operands: 82
81	Operation	operator: 83 operands: 84
82	ExprTuple	86, 85
83	Literal
84	ExprTuple	86, 87
85	Literal
86	Variable
87	Variable